The Foundation of Computing

Every digital device—from your smartphone to supercomputers—operates fundamentally on binary code. The binary numeral system (Base-2) uses only two symbols: 0 and 1. In physical hardware, a 1 represents a microscopic transistor being "ON" (allowing electricity to flow), while a 0 represents it being "OFF." By stringing millions of these binary switches together, computers can perform complex arithmetic and logic instantly.

Arithmetic Operations

Binary arithmetic works identically to standard decimal (Base-10) math, but with fewer digits.

• Addition (+): 0+0=0, 1+0=1, 0+1=1. The only trick is 1+1=10 (Write down 0, carry the 1 to the next column).
• Subtraction (-): Works exactly like decimal subtraction, including "borrowing" from the next column if you try to subtract 1 from 0.
• Multiplication (×): Extremely simple in binary. You multiply by 1 (copy the number) or multiply by 0 (write zeros). Then add the staggered results together.

Bitwise Logic Operations

Instead of doing arithmetic, computers often use Logic Gates to compare binary numbers bit-by-bit. This is the core of computer programming and networking (like subnet masking).

• AND: Compares two bits. The result is 1 only if both bits are 1. Otherwise, it is 0.
• OR: The result is 1 if either (or both) bits are 1.
• XOR (Exclusive OR): The result is 1 only if the bits are different (one is 1, the other is 0). If they match, the result is 0. Used heavily in cryptography.
• NOT: A unary operation that simply flips the bits. Every 1 becomes a 0, and every 0 becomes a 1.

Bit Shifting

Bit shifting literally moves the entire binary string left or right.

• Left Shift (<<): Moves bits to the left, appending a 0 to the right. Mathematically, this is a highly efficient way to multiply an integer by 2. (e.g., 0011 [3 in decimal] shifted left becomes 0110 [6 in decimal]).
• Right Shift (>>): Moves bits to the right, discarding the final bit. This effectively divides an integer by 2, dropping any remainder.